Natural question is what happens to the anomaly cancellation conditions
at a non-perturbative level and if some symmetry is only approximate. The two issues are indeed related;
non-perturbative results are more accessible in presence of finite cut-offs but some of the symmetries
are broken. The non-perturbative construction of electroweak theory including the femionic sector
in the limit of removed cut-off
(if possible at all) faces two major and unsolved problems:
first the theory is not asymptotically free, and second there is no a suitable,
lattice regularization for chiral gauge theory
[1]}, [2]}, [3]},[4]}( in QED only the
first problem is present). It is however not really necessary to remove the ultraviolet cut-off
to have a physically meaningful theory; one can fix the cut-off and consider the Standard electroweak
as an
Effective Quantum Field Theory [5]},[6]},[7]} valid up to a certain energy scale, reliable for predictions at energy scales far from the cut-off.
Also the symmetry breaking due to finite cut-off has a physical meaning, as some of the symmetries could be indeed only emerging
and broken at a more fundamental level.
Both non-perturbative effects or symmetry violations could affect the conservation of hypercharge current under the
anomaly cancellation conditions [8]}.
